Bezdek's axial-symmetry conjecture for plane sections of convex bodies

Let $K\subset \mathbb{R}^3$ be a convex body. A plane section means the intersection of $K$ with an affine plane, and an axis of symmetry is a line in that plane whose reflection preserves the section. Bezdek's conjecture. If all plane sections of $K$ have an axis of symmetry, then $K$ is an ellipsoid or a body of revolution. This conjecture asks whether axial symmetry of every plane section forces one of these global forms; the supplied source gives no resolution status.